In another of Cersei's half-assed measures, she kills the High Septon Tyrion elevated without making sure her own creature would wear the crystal crown. That combined with her overlooking of the sparrow movement due to her own class prejudices resulted in the replacement of a Lannister toady with one hostile to her regime. Worse, she gave him an army in another act of carelessness and shortsightedness.
She may have another chance when the High Septon goes south to support Aegon. The war is going to affect the office of the High Septon itself as different contenders for the Iron Throne will put forward their own candidates.
The Holy Roman emperors had their share of conflicts with the papacy. They would support antipopes in their conflicts with the Church the popes could also play this game by supporting anti-kings. The Iron Throne is little different as the monarchy has been interfering with the Faith since Maegor the Cruel named his own candidate for High Septon when he was warring with the Faith. Jaehaerys I followed up having learned the dangers of having a High Septon who opposes the monarchy, but using bribery rather than force. Cersei would likely pick someone to take the High Sparrow's place in Baelor's Sept to give the Lannister regime their divine support.
"Do not discount this man Luceon," Qyburn said. "Last night he feted thirty of the Most Devout on suckling pig and Arbor gold, and by day he hands out hardbread to the poor to prove his piety."
-AFFC, Cersei IV
Qyburn's whisperers claimed that Septon Luceon had been nine votes from elevation when those doors had given way, and the sparrows came pouring into the Great Sept with their leader on their shoulders and their axes in their hands.
-AFFC, Cersei VI
Septon Luceon is the most likely candidate as he was runner-up before the High Sparrow took control. He is a foil as while also being from the riverlands like the High Sparrow, he is a nobleman, being a member of House Frey which is allied with House Lannister. The High Sparrow was a traveling septon who catered to the smallfolk while Septon Luceon is an ecclesiastical careerist who went to Baelor's Sept to work his way up the Faith hierarchy. While the High Sparrow is an admitted self-flagellate, Septon Luceon lives it up.
His acts of piety are purely PR, serving bread to the poor while serving luxury cuisine to the Most Devout to buy votes. It's not even fresh bread, but bread that has gone stale, the cheapest fare, demonstrating his clear contempt for the poor in spite of the values he preaches.
Most appealingly, he is corrupt, and Cersei is following Tywin's modus operandi of picking the most opportunistic and bribable actors given as Tywin said "who can bid higher than us?" Of course, that overlooks the long-term consequences of her regime being associated with corruption and uninhibited self-aggrandizement and opportunism.
History rhymes apparently as Septon Luceon will be her anti-High Septon to challenge the High Sparrow as the Lannisters war with the Faith. He would be calling for peace, dissolution of the Faith Militant and an end to the fighting while saying nothing of the Lannisters. He would have no following among the smallfolk, but Septon Luceon may instead be given the same derision as Maegor's "High Lickpsittle" for being Cersei's puppet.
The faithful throughout the realm will have their choices between a corrupt puppet and a power hungry zealot as their High Septon. It seems like a bleak choice between two extremes on the spectrum. The High Sparrow is commonborn but has the Faith Militant while the High Lickpsittle is highborn, but will likely have the gold, especially with Casterly Rock backing him, giving it a class dimension.
Not to mention that their respective factions are mirror images of each other: campaigns to seat imposter boy kings that carry an association with gold, mistreatment of the smallfolk, infanticide and not to mention Connington wanting to be Tywin 2.0.
Stannis has no candidate for High Septon given he joined the Red Faith. As for Daenerys, however, it's an open question if she puts forward her own candidate. The High Sparrow will oppose her for being a mix of all the things he hates about the other players: she's a woman ("all women are wantons at heart, given to using their wiles and their beauty to work their wills on men"), she's backed by the Red Faith ("red demon") and she's opposing his royal candidate. The claim that he crowned an imposter would be politically embarrassing for him. The High Lickpsittle will oppose Daenerys, because Cersei does. I wouldn't rule out some misogyny either. It'd be interesting to see who Daenerys picks for her candidate if she does.
As to who is High Septon in the end, I don't know, but I think it may be a character we already met. It could be Septon Meribald.
He's a commonborn traveling septon who cares for the smallfolk like the High Sparrow but without his worse traits of zealotry, hunger for power and intense misogyny. He would also have Septon Luceon's desire for peace, but without his corruption and being a puppet.
Note: He could even "tame" Nymeria by offering her food from his mule and/or perhaps pulling an arrow from her or healing a wound, performing what would be seen as a genuine miracle a la St. Francis of Assisi and the Wolf of Gubbio. He manages to neutralize a threat to the people of the riverlands not through the sword but compassion.
He would call for a dissolution of the Faith Militant, but out of his experience as both a war veteran and seeing its effects on the smallfolk. Meribald's attitude would be that there are values that cannot be expressed through violence, and that starvation can often be as great if not greater threat to the smallfolk now that winter has come. He would more likely send them carting food to the smallfolk throughout the realm during winter.
